Solving Integrals from Hell for n = 1, 1/2, 2, 3/2

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In summary, the conversation revolved around solving a set of integrals with multiple parameters, including n, ω, φ, and λ. The question was whether these integrals could be solved analytically or if approximations would be necessary. It was mentioned that there are no restrictions on the parameters, but setting some may be helpful in avoiding the need for approximations. The origin of the integrals was also discussed. Ultimately, there were no definitive answers on the restrictions or analytical solutions for the integrals.
  • #1
l'Hôpital
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Hi, I was just wondering if these integrals could be solved analytically, or if I would just have to resort to approximations.

[tex]
\int_{0}^{\infty} \sqrt{1 + \omega E^2} E^n ln(1 + \omega E^2) \frac{e^{\phi E}}{(\lambda e^{\phi E} + 1)^2} dE
[/tex]

For

[tex]
n = 1, 1/2, 2, 3/2
[/tex]
 
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  • #2
any restrictions on any parameters? For example, is it true that [tex]\omega[/tex] is real and positive? Is [tex]\phi[/tex] real? Any restrictions on [tex]\lambda[/tex]?
 
  • #3
No restrictions. But if you'd like to set some, go for it.

I'd just rather avoid approximation unless they wouldn't perturb the value by much.
 
  • #4
Surely there must be some restrictions. For example, if [tex] \lambda = 0 [/tex] then the real part of [tex] \phi [/tex] must be negative, otherwise the integral doesn't converge. Are there any special cases that are particularly interesting?

I am playing with it, but don't have my hopes up.

By the way, where did this integral come from? It is unusual to have so many arbitrary parameters (meaning all can be arbitrary complex numbers) in the practical problems I usually run into. jason
 
  • #5
Truth be told, I don't know if there are any restrictions.

A friend asked me for my opinion on these integrals, if there was a way to solve them analytically. Of course, I was stumped, so I figured I'd do him the favor of posting these up in PF.

I'll try to see if I can find some restrictions on them.
 

FAQ: Solving Integrals from Hell for n = 1, 1/2, 2, 3/2

What is an integral?

An integral is a mathematical concept that represents the area under a curve in a graph. It is used to calculate the total value of a function over a given interval.

What is the significance of n in "Solving Integrals from Hell for n = 1, 1/2, 2, 3/2"?

In this context, n represents the power or exponent of the function that is being integrated. It helps to determine the complexity of the integral and the methods that can be used to solve it.

What makes solving integrals for n = 1, 1/2, 2, 3/2 challenging?

Integrals with non-integer values of n, such as 1/2 and 3/2, are known as fractional integrals and can be more difficult to solve compared to integrals with integer values of n. This is because they require more advanced techniques and may not have closed-form solutions.

How can integrals for n = 1, 1/2, 2, 3/2 be solved?

There are various techniques that can be used to solve integrals for n = 1, 1/2, 2, 3/2, such as substitution, integration by parts, and trigonometric substitutions. It is important to choose the appropriate technique based on the specific integral and the value of n.

What are some real-world applications of solving integrals for n = 1, 1/2, 2, 3/2?

Solving integrals for n = 1, 1/2, 2, 3/2 has various applications in physics, engineering, and other fields. For example, in physics, integrals are used to calculate the work done by a force, the displacement of an object, or the amount of energy in a system. In engineering, integrals are used to calculate the area under a stress-strain curve to determine the strength of a material. They are also used in economics, biology, and many other areas.

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